binomial theorem 2012

South Western Sydney Region HSC Mathematics Extension 1-2 Study DayBankstown Senior College, September 2012

THE BINOMIAL THEOREMRobert Yen

Outline1. Introduction 6. Finding the greatest coefficient2. Binomial expansions and Pascal’s triangle 7. Proving identities involving the sum of 3. nCk, a formula for Pascal’s triangle coefficients nCk

4. The binomial theorem in the past 10 HSC exams 8. Binomial probability5. Finding a particular term 9. How to study for Maths: a 4-step approach

1. INTRODUCTION The PowerPoint presentation for these notes can be found at HSC Online:

http://hsc.csu.edu.au/maths/ext1/binomial_theorem/ This topic examines the general pattern for expanding (a + x)n

It is a difficult topic because it involves new work on high-level algebra and is learned at the end of the course with little time for practice and revision

HSC questions involving this topic are often targeted at better Extension 1 students, especially when they appear in Question 7, so if you are aiming to achieve at the highest band (E4) in this course, work on mastering this topic to excel in the exam

There are no shortcuts to success in this topic: you just have to learn the theory to develop a full understanding

2. BINOMIAL EXPANSIONS AND PASCAL’S TRIANGLEBinomial expansion No. of terms Coefficients of terms

(a + x)1 = a + x 2 1 1(a + x)2 = a2 + 2ax + x2 3 1 2 1(a + x)3 = a3 + 3a2x + 3ax2 + x3 4 1 3 3 1(a + x)4 = a4 + 4a3x + 6a2x2 + 4ax3 + x4 5 1 4 6 4 1(a + x)5 = a5 + 5a4x + 10a3x2 + 10a2x3 + 5ax4 + x5 6 1 5 10 10 5 1

(a + x)n has n + 1 terms, with the powers of a decreasing from n to 0 and the powers of x increasing from 0 to n.

The sum of the powers in each term is always n. The coefficients of the terms appear in Pascal’s triangle.

3. nCk, A FORMULA FOR PASCAL’S TRIANGLE nCk from the Permutations and combinations topic also gives the value of row n, term k of

Pascal’s triangle, if we start counting from row 0, term 0

1 0C0

1 1 1C0 1C1

1 2 1 2C0 2C1

2C2

1 3 3 1 3C0 3C1

3C2 3C3

1 4 6 4 1 4C0 4C1

4C2 4C3

4C4

1 5 10 10 5 1 5C0 5C1

5C2 5C3

5C4 5C5

1 6 15 20 15 6 1 6C0 6C1

6C2 6C3

6C4 6C5

6C6

1 7 21 35 35 21 7 1 7C0 7C1

7C2 7C3

7C4 7C5

7C6 7C7

1 8 28 56 70 56 28 8 1 8C0 8C1

8C2 8C3

8C4 8C5

8C6 8C7

8C8

C stands for coefficient as well as combination, and nCk is also written as

There are 3 ways of calculating 5C3:

(a) Mentally: 5C3 = = 10

(b) Formula: 5C3 = = = 10, using nCk =

This works for 5C3 because .

(c) Calculator key: pressing 5 nCr 3 = gives 10.

The binomial theorem

(a + x)n = nC0 an + nC1 an-1 x + nC2 an-2 x2 + nC3 an-3 x3 + nC4 an-4 x4 + … + nCn xn

or in sigma notation:

(a + x)n =

the sum of terms the general termfrom k = 0 to n

Properties of nCk

1. nC0 = nCn = 1 First and last coefficients are 1

2. nC1 = nCn-1 = n Second and second-last coefficients are n

3. nCk = nCn-k Pascal’s triangle is symmetrical, for example, 6C2 = 6C4

4. n+1Ck = nCk-1 + nCk Pascal’s triangle result: each coefficient is the sum of the two coefficients in the row above it

Example 1Use the binomial theorem to expand: Answers

(a) (a + 3)5 a5 + 15a4 + 90a3 + 270a2 + 405a + 243

(b) (2x – y)4 16x4 – 32x3y + 24x2y2 – 8xy3 + y4

The binomial theorem: Robert Yen (page 2)

nCk is the coefficient in the term that

contains xk in the expansion

4. THE BINOMIAL THEOREM IN THE PAST 10 HSC EXAMS

HSCexam

Finding a particular term

Finding the greatest

coefficient

Proving identities

Binomialprobability

2002 last tested in 1988, Q6(b)

Q7(b) Q4(a)2003 Q2(d) Q3(c), Q8(a) Ext 22004 Q7(b) Q4(c)2005 Q2(b) Q6(a)2006 Q2(b) Q6(b)2007 Q6(a)(i) Ext 2 Q4(a)2008 Q1(d), Q6(c)(i) Q6(c)(ii),

Q6(c) Ext 22009 Q6(b) Q4(a)2010 Q7(b) Q1(f)2012 Q2(c) Q7(b) Q6(c)

5. FINDING A PARTICULAR TERM

Example 2 (2008 HSC, Question 1(d), 2 marks)Find an expression for the coefficient of x8y4 in the expansion of (2x + 3y)12.

[Answer: 12C4 28 34]

Steps for finding a particular term

1. Write a formula for the general term Tk of the expansion and simplify the formula,for example, Tk = 12Ck (2x)12-k (3y)k.

2. To find the term with the required power of x, solve an equation for k, for example, 12 – k = 8, or k = 4.

k must be a whole number or you have made a mistake.

3. Substitute the value of k into the Tk formula to find the required term.

Tk is not the kth term! In the expansion of (a + x)n, Tk is the term that contains xk

It is not the kth term but actually the (k + 1)th term, for example, T3 is the 4th term (T0, T1, T2,

T3), the one that contains x3

It is simpler to write out the first few terms of the expansion rather than try to memorise the sigma notation

It is also better to avoid referring to the ‘kth term’ and calling its formula Tk+1 (as some textbooks do) because students can get confused about the value of k to substitute (in Example 2 above, some substituted k = 5 ‘for the 5th term’ instead of k = 4)

Anyway, HSC questions will tell you to find, for example, ‘the term that contains x8’ rather than ‘the 9th term’

Example 3 (2011 HSC, Question 2(c), 2 marks)

Find an expression for the coefficient of x2 in the expansion of .

[Answer: -870 912 or 8C3 35 (-4)3]


Common student mistakes Giving the position of the term (‘the 5th term’) rather than the actual term Poor use of algebra, index laws, brackets and negative signs Wasting time expanding out all the terms Substituting wrong value for k, such as k + 1 instead In Example 2, giving the coefficient as 12C4 only instead of 12C4 (28)(34)

6. FINDING THE GREATEST COEFFICIENTIn (1 + 2x)8 = 1 + 16x + 112x2 + 448x3 + 1120x4 + 1792x5 + 1792x6 + 1024x7 + 256x8, the

greatest coefficient is 1792 (occurring twice).The term with the greatest coefficient usually occurs in the middle of an expansion because

with the rows in Pascal’s triangle, the larger numbers are in the middle. In any expansion of (a + x)n, the coefficients usually increase, reach a maximum, then decrease.

Example 4

Suppose (1 + 2x)8 = .

(a) Find an expression for tk, the coefficient of xk. [Answer: 8Ck 2k]

(b) Show that .

(c) Show that the greatest coefficient is 1792.

Steps for finding the greatest coefficient

1. Write formulas for the general coefficient tk and the next coefficient tk+1.

2. Simplify to an expression of the form .

Note that = n and = c1 = c.

3. Solve > 1 to find the highest integer value of k.

4. Find the value of tk+1, the greatest coefficient.

HOMEWORK EXERCISE (1988 HSC, Question 6(b), 6 marks)

Suppose (7 + 3x)25 = .

(i) Use the binomial theorem to write an expression for tk, 0 k 25.

(ii) Show that .

(iii) Hence or otherwise find the largest coefficient tk.

You may leave your answer in the form .

[Answer: (≈ 1.71 × 10 )]


7. PROVING IDENTITIES INVOLVING THE SUM OF COEFFICIENTS nCk

Pascal’s triangle n ΣnCksum of values

Σ(nCk)2

sum of (values)2

1 0 1 (20) 1 (1C0)1 1 1 2 (21) 2 (2C1)

1 2 1 2 4 (22) 6 (4C2)1 3 3 1 3 8 (23) 20 (6C3)

1 4 6 4 1 4 16 (24) 70 (8C4)1 5 10 10 5 1 5 32 (25) 252 (10C5)

1 6 15 20 15 6 1 6 64 (26) 924 (12C6)1 7 21 35 35 21 7 1 7 128 (27) 3432 (14C7)

1 8 28 56 70 56 28 8 1 8 256 (28) 12 870 (16C8)

Two important identities:

1.

For example, .

2.

For example, .

Identities involving the sum of coefficients can be proved by expanding (1 x)n and then: substituting x = 0, 1 or -1, or equating coefficients, or differentiating or integrating.

The binomial theorem for (1 + x)n

(1 + x)n = nC0 + nC1 x + nC2 x2 + nC3 x3 + nC4 x4 + … + nCn xn

or in sigma notation: (1 + x)n =

Example 5 (similar to 2010 HSC, Question 7(b)(i), 1 mark)

Expand (1 + x)n and substitute an appropriate value of x to prove that .

Example 6By considering that (1 + x)2n = (1 + x)n(1 + x)n and examining the coefficient of xn on each side,

prove that .


Hints for proving identities (by John Dillon, Head Teacher of Maths, Hurlstone AHS)

(1 + x)n = nC0 + nC1 x + nC2 x2 + nC3 x3 + …+ nCn xn

If the identity involves ... try ...nCk’s with no x’s substituting a simple value such as x = 0 or x =

1nCk’s with alternating + and – signs substituting a negative value for xpowers of a number (say a) as well as nCk’s substituting x = anCk’s multiplied by k’s differentiating both sidesnCk’s divided by (k + 1)’s integrating both sides but don’t forget ‘+ c’

HOMEWORK EXERCISES1 Expand both sides of the identity (1 + x)n(1 + x) = (1 + x)n+1 and compare coefficients to prove

Pascal’s triangle result n+1Ck = nCk-1 + nCk.2 (2010 HSC, Question 7(b)(iii), 2 marks)

Expand (1 + x)n and differentiate both sides to prove that .

Example 7 (2006 HSC, Question 2(b), 2 marks) (i) By applying the binomial theorem to (1 + x)n and differentiating, show that 1

(ii) Hence deduce that 1

Example 8 (2008 HSC, Question 6(c), 5 marks)Let p and q be positive integers with p q.

(i) Use the binomial theorem to expand (1 + x)p + q, and hence write down the term 2

of which is independent of x. [Answer to (i) and (ii): ]

(ii) Given that , apply the binomial theorem and the result of 3

part (i) to find a simpler expression for 1 + .

Common student mistakes Messy and careless working, unclear notation; not enough working, ‘fudging’ the answer Forgetting the first term nC0 or the last term nCn xn

Using series formulas or mathematical induction instead of the binomial theorem: this usually doesn’t work

Not realising that the parts of the question are related Getting lost in sigma notation; from the examiners’ notes on the 2008 HSC exam (p.6):

‘Responses that used sigma notation were sometimes less successful than (students) who wrote out the sum showing at least three correct terms. Many ... misinterpreted this part of the question by stating which term was independent of x rather than by giving the independent term or, by being careless in their notation, failed to gain this mark.’

If integrating, forgetting the constant at the end


Starting a proof using the identity to be proved, rather than prove that LHS = RHS


Example 9 (2002 HSC, Question 7(b), 6 marks, HARD!)The coefficient of xk in (1 + x)n, where n is a positive integer, is denoted by ck (so ck = nCk).

(i) Show that c0 + 2c1 + 3c2 + … + (n + 1)cn = (n + 2) 2n-1. 3

(ii) Find the sum . 3

Write your answer as a simple expression in terms of n. [Answer: ]

8. BINOMIAL PROBABILITY (for you to study at home)With binomial probability, we are concerned with repeated trials in which there are only two

possible outcomes: we can call one outcome a success, with probability p, and the other outcome a failure, with probability q = 1 – p. Examples of such outcomes are heads vs. tails, win vs. lose, true vs. false, boy vs. girl, defective vs. working.

If a binomial trial is repeated n times, then the probability of r successes isP(X = r) = nCr pr qn-r

X is called the random variable and its value ranges from 0 to n.

Example 10 (2007 HSC, Question 4(a), 4 marks)In a large city, 10% of the population has green eyes.

(i) What is the probability that two randomly chosen people both have green eyes? 1[Answer: 0.01]

(ii) What is the probability that exactly two of a group of 20 randomly chosen people 1have green eyes? Give your answer correct to three decimal places.

[Answer: 0.285](iii) What is the probability that more than two of a group of 20 randomly chosen people 2

have green eyes? Give your answer correct to two decimal places.[Answer: 0.32]

Note that this is an application of the binomial theorem, because:P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + ... + P(X = 20) = 20C0 0.10 0.920 + 20C1 0.11 0.919 + 20C2 0.12 0.918 + 20C3 0.13 0.917 + …+ 20C20 0.1200.90

= (0.9 + 0.1)20 that is, (q + p)n = 120 = 1.

Common student mistakes Not using the complementary result as a shortcut Forgetting to include P(X = 0) Does ‘more than two’ include two?


Each probability is a term of the expansion

of (q + p)nThe sum of the probabilities of all possible events is 1

HOMEWORK EXERCISE (2004 HSC, Question 4(c))Katie is one of ten members of a social club. Each week one member is selected at random to

win a prize.

(i) What is the probability that in the first 7 weeks Katie will win at least 1 prize? 1

[Answer: 1 – ]

(ii) Show that in the first 20 weeks Katie has a greater chance of winning exactly 2 prizes 2 than of winning exactly 1 prize.

[Answer: P(X = 2) ≈ 0.2852 > P(X = 1) ≈ 0.2702](iii) For how many weeks must Katie participate in the prize drawing so that she has 2 a greater chance of winning exactly 3 prizes than of winning exactly 2 prizes?

[Answer: 30 weeks]

9. HOW TO STUDY FOR MATHS: A 4-STEP APPROACH (P-R-A-C)

1. PRACTISE YOUR MATHS Master your skills, strengthen your ability Achieve a high level of understanding

2. REWRITE YOUR MATHS Summarise the theory and examples in your own words Work through all topics to see the big picture Achieve an overview of the whole course

3. ATTACK YOUR MATHS Identify your areas of weakness and work on overcoming them Fill in any gaps in your mathematical knowledge

4. CHECK YOUR MATHS Revise your understanding on mixed revision exercises and past HSC exams

Before an exam Review and memorise your topic summaries Practise on your weak areas Practise on HSC-style questions Anticipate the exam: the format and structure, the style of questions, planning your time

during the exam. Useful resources

NSW HSC Online has tips, tutorials and links: http://hsc.csu.edu.au/maths/ The Board of Studies website has past HSC exams, sample solutions, marking

guidelines and markers’ comments: www.boardofstudies.nsw.edu.au/hsc_exams The HSC Advice Line provides access for HSC students to experienced teachers in 20

HSC subjects, starting in October: call 13 11 12 from a fixed-line phone for the cost of a local call (30 cents)

The Mathematical Association of NSW sells booklets of past HSC exams with worked solutions; you may be able to buy these through your school: www.mansw.nsw.edu.au


binomial theorem 2012

Documents